Question

Please Use R programming language to answers these question and please show me the code as well. Thank You

1. Problem: dataset: savings; package : faraway
Use R, perform the calculations and answer the following questions
(a) Calculate the design matrix X, and all regression coefficients estimates, as shown in (3).
(b) Calculate the Residuals standard error , as in (5).
(c) ANOVA table: Calculate SST, SSE, SSR, ?2, as in (6).
     Calculate the ANOVA F-statistic and p-value.
(d) Build a reduced model. Which predictors you take out? Why? (8)
     Use anova() to compare 2 models. Conclusions.
(e) Perform 1 prediction.

Answer 1

(a)

library(faraway)
data(savings)
X=savings
RM=lm(sr ~ pop15 + pop75 + dpi + ddpi, savings)
RM

Call:
lm(formula = sr ~ pop15 + pop75 + dpi + ddpi, data = savings)

Coefficients:
(Intercept)          pop15            pop75             dpi                ddpi
28.5660865   -0.4611931   -1.6914977   -0.0003369    0.4096949

(b)

summary(RM)

Call:
lm(formula = sr ~ pop15 + pop75 + dpi + ddpi, data = savings)

Residuals:
    Min      1Q Median      3Q     Max
-8.2422 -2.6857 -0.2488 2.4280 9.7509

Coefficients:
                     Estimate    Std. Error    t value Pr(>|t|)  
(Intercept) 28.5660865 7.3545161   3.884 0.000334 ***
pop15       -0.4611931   0.1446422 -3.189 0.002603 **
pop75       -1.6914977 1.0835989 -1.561 0.125530  
dpi            -0.000336    0.0009311 -0.362 0.719173  
ddpi         0.4096949    0.1961971   2.088 0.042471 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.803 on 45 degrees of freedom
Multiple R-squared: 0.3385,    Adjusted R-squared: 0.2797
F-statistic: 5.756 on 4 and 45 DF, p-value: 0.0007904

The Residuals standard error is 3.803

(c)

anova(RM)

Analysis of Variance Table

Response: sr
                  Df Sum Sq Mean Sq F value    Pr(>F)  
pop15        1   204.12   204.118   14.1157 0.0004922 ***
pop75        1   53.34     53.343     3.6889   0.0611255 .
dpi             1   12.40     12.401     0.8576   0.3593551  
ddpi           1   63.05     63.054     4.3605   0.0424711 *
Residuals 45 650.71    14.460                    
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(d) As we see in the previous result that pop15 and ddpi significantly effect the savings rate. So, we ignoring the other insignificantly variable, do the analysis

RM1=lm(sr ~ pop15 + ddpi, savings)
summary(RM1)

Call:
lm(formula = sr ~ pop15 + ddpi, data = savings)

Residuals:
    Min      1Q Median      3Q     Max
-7.5831 -2.8632 0.0453 2.2273 10.4753

Coefficients:
            Estimate Std. Error t value Pr(>|t|)  
(Intercept) 15.59958    2.33439   6.682 2.48e-08 ***
pop15       -0.21638    0.06033 -3.586 0.000796 ***
ddpi         0.44283    0.19240   2.302 0.025837 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.861 on 47 degrees of freedom
Multiple R-squared: 0.2878,    Adjusted R-squared: 0.2575
F-statistic: 9.496 on 2 and 47 DF, p-value: 0.0003438

anova(RM1)

Analysis of Variance Table

Response: sr
          Df Sum Sq Mean Sq F value    Pr(>F)  
pop15      1 204.12 204.118 13.6942 0.0005633 ***
ddpi       1 78.96 78.959 5.2973 0.0258374 *
Residuals 47 700.55 14.905                    
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1